L(s) = 1 | + (−5.03 − 2.57i)2-s + (3.80 + 19.1i)3-s + (18.7 + 25.9i)4-s + (8.73 − 13.0i)5-s + (30.1 − 106. i)6-s + (126. − 84.6i)7-s + (−27.4 − 178. i)8-s + (−127. + 52.8i)9-s + (−77.6 + 43.3i)10-s + (307. + 61.2i)11-s + (−425. + 457. i)12-s + (−241. + 241. i)13-s + (−856. + 99.9i)14-s + (283. + 117. i)15-s + (−322. + 971. i)16-s + (1.04e3 + 570. i)17-s + ⋯ |
L(s) = 1 | + (−0.890 − 0.455i)2-s + (0.244 + 1.22i)3-s + (0.585 + 0.810i)4-s + (0.156 − 0.233i)5-s + (0.341 − 1.20i)6-s + (0.977 − 0.652i)7-s + (−0.151 − 0.988i)8-s + (−0.524 + 0.217i)9-s + (−0.245 + 0.136i)10-s + (0.766 + 0.152i)11-s + (−0.852 + 0.916i)12-s + (−0.395 + 0.395i)13-s + (−1.16 + 0.136i)14-s + (0.325 + 0.134i)15-s + (−0.315 + 0.949i)16-s + (0.878 + 0.478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.39275 + 0.528386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39275 + 0.528386i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.03 + 2.57i)T \) |
| 17 | \( 1 + (-1.04e3 - 570. i)T \) |
good | 3 | \( 1 + (-3.80 - 19.1i)T + (-224. + 92.9i)T^{2} \) |
| 5 | \( 1 + (-8.73 + 13.0i)T + (-1.19e3 - 2.88e3i)T^{2} \) |
| 7 | \( 1 + (-126. + 84.6i)T + (6.43e3 - 1.55e4i)T^{2} \) |
| 11 | \( 1 + (-307. - 61.2i)T + (1.48e5 + 6.16e4i)T^{2} \) |
| 13 | \( 1 + (241. - 241. i)T - 3.71e5iT^{2} \) |
| 19 | \( 1 + (-208. + 502. i)T + (-1.75e6 - 1.75e6i)T^{2} \) |
| 23 | \( 1 + (-185. + 932. i)T + (-5.94e6 - 2.46e6i)T^{2} \) |
| 29 | \( 1 + (-4.15e3 - 2.77e3i)T + (7.84e6 + 1.89e7i)T^{2} \) |
| 31 | \( 1 + (3.72e3 - 741. i)T + (2.64e7 - 1.09e7i)T^{2} \) |
| 37 | \( 1 + (9.47e3 - 1.88e3i)T + (6.40e7 - 2.65e7i)T^{2} \) |
| 41 | \( 1 + (-1.37e3 - 2.06e3i)T + (-4.43e7 + 1.07e8i)T^{2} \) |
| 43 | \( 1 + (-1.78e3 - 4.30e3i)T + (-1.03e8 + 1.03e8i)T^{2} \) |
| 47 | \( 1 + (3.56e3 + 3.56e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (3.71e3 + 1.53e3i)T + (2.95e8 + 2.95e8i)T^{2} \) |
| 59 | \( 1 + (-4.71e4 + 1.95e4i)T + (5.05e8 - 5.05e8i)T^{2} \) |
| 61 | \( 1 + (2.67e4 - 1.78e4i)T + (3.23e8 - 7.80e8i)T^{2} \) |
| 67 | \( 1 - 6.16e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.44e4 + 7.27e4i)T + (-1.66e9 + 6.90e8i)T^{2} \) |
| 73 | \( 1 + (-1.77e4 + 2.65e4i)T + (-7.93e8 - 1.91e9i)T^{2} \) |
| 79 | \( 1 + (8.19e3 + 1.62e3i)T + (2.84e9 + 1.17e9i)T^{2} \) |
| 83 | \( 1 + (5.67e4 + 2.35e4i)T + (2.78e9 + 2.78e9i)T^{2} \) |
| 89 | \( 1 + (-5.69e4 - 5.69e4i)T + 5.58e9iT^{2} \) |
| 97 | \( 1 + (1.44e5 + 9.68e4i)T + (3.28e9 + 7.93e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.19195695187540250238017933506, −12.48131323376117681433699976412, −11.26292788400780255777466230978, −10.37577021824603122644679684087, −9.447568734546507013545163228167, −8.483318265959841107615724846986, −7.08778307867587432334169999973, −4.77959817137982079607725714850, −3.54713622108512946845812300253, −1.40253499186842185114316226947,
1.07529387188777331214697020372, 2.32469142724728196670234576412, 5.45420409840489497255398464154, 6.76456601536783031060903472191, 7.79003004062887357020724998195, 8.643019859466549192650195903508, 10.05064652924345877210108532861, 11.52481110069430171161301401367, 12.33766393112519009630104613287, 14.05173837990018867871731678554